With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bipartition of a multiqubit system over the possible sequence of measurements of random unknown states and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multiqubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences, which allows one to deal naturally with incomplete information, i.e., information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number increases only like the third power of the number $N$ of qubits. Their efficiency increases rapidly with $N$, leaving already for $N=4$ less than a fraction ${10}^{-6}$ of randomly chosen entangled states undetected.

• Received 25 February 2019

DOI:https://doi.org/10.1103/PhysRevA.100.012328